Problem: Integrate. $ \int 5\csc^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-10\csc^2(x)\cot(x)+C$ (Choice B) B $-5\csc^2(x)\cot(x)+C$ (Choice C) C $-5\cot(x)+C$ (Choice D) D $-\cot(x)+C$
Solution: We need a function whose derivative is $5\csc^2(x)$. We know that the derivative of $\cot(x)$ is $-\csc^2(x)$, so let's start there: $\dfrac{d}{dx} \cot(x) = -\csc^2(x)$ Now let's multiply by $-5$ : $\dfrac{d}{dx} \left[ -5\cot(x) \right]= -5\dfrac{d}{dx} \cot(x) = 5\csc^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 5\csc^2(x)\,dx =-5 \cot(x)\, + C$ The answer: $-5 \cot(x)\, + C$